Subgroups — C₂×C₄²×C₅×C₉×C₁₇ #72201750

Order 24480.

648 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₉×C₁₇	C₂	1	3	yes	maximal
2	C₄²×C₅×C₉×C₁₇	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₄²×C₅×C₁₇	C₃	1	1	yes	maximal, characteristic
4	C₂³×C₅×C₉×C₁₇	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₉×C₁₇	C₄	1	12	yes	direct product factor
6	C₂×C₄×C₅×C₉×C₁₇	C₂²	1	6	yes	—
7	C₂×C₄²×C₉×C₁₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₄×C₅×C₁₇	C₂×C₃	1	3	yes	—
9	C₃×C₄²×C₅×C₁₇	C₂×C₃	1	4	yes	—
10	C₂×C₄²×C₅×C₁₇	C₉	1	1	yes	characteristic, direct product factor, Hall
11	C₂²×C₄×C₉×C₁₇	C₂×C₅	1	3	yes	—
12	C₄²×C₉×C₁₇	C₂×C₅	1	4	yes	direct product factor
13	C₂³×C₃×C₅×C₁₇	C₂²×C₃	1	1	yes	characteristic
14	C₂×C₃×C₄×C₅×C₁₇	C₃×C₄	1	12	yes	—
15	C₂×C₃×C₄×C₅×C₁₇	C₂²×C₃	1	6	yes	—
16	C₂²×C₅×C₉×C₁₇	C₂×C₄	1	6	yes	—
17	C₂²×C₅×C₉×C₁₇	C₂³	1	1	yes	characteristic
18	C₄×C₅×C₉×C₁₇	C₂×C₄	1	12	yes	direct product factor
19	C₂×C₃×C₄²×C₁₇	C₃×C₅	1	1	yes	characteristic
20	C₂×C₅×C₉×C₁₇	C₄²	1	4	yes	direct product factor
21	C₂×C₅×C₉×C₁₇	C₂²×C₄	1	3	yes	—
22	C₂×C₄²×C₅×C₉	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂²×C₄×C₅×C₁₇	C₂×C₉	1	3	yes	—
24	C₄²×C₅×C₁₇	C₂×C₉	1	4	yes	direct product factor
25	C₂³×C₉×C₁₇	C₂²×C₅	1	1	yes	characteristic
26	C₂×C₄×C₉×C₁₇	C₄×C₅	1	12	yes	direct product factor
27	C₂×C₄×C₉×C₁₇	C₂²×C₅	1	6	yes	—
28	C₂²×C₃×C₅×C₁₇	C₂×C₃×C₄	1	6	yes	—
29	C₂²×C₃×C₅×C₁₇	C₂³×C₃	1	1	yes	characteristic
30	C₃×C₄×C₅×C₁₇	C₂×C₃×C₄	1	12	yes	—
31	C₂²×C₃×C₄×C₁₇	C₂×C₃×C₅	1	3	yes	—
32	C₃×C₄²×C₁₇	C₂×C₃×C₅	1	4	yes	—
33	C₅×C₉×C₁₇	C₂×C₄²	1	1	yes	characteristic, direct product factor, Hall
34	C₂²×C₄×C₅×C₉	C₂×C₁₇	1	3	yes	—
35	C₄²×C₅×C₉	C₂×C₁₇	1	4	yes	direct product factor
36	C₂³×C₅×C₁₇	C₂²×C₉	1	1	yes	characteristic
37	C₂×C₄×C₅×C₁₇	C₄×C₉	1	12	yes	direct product factor
38	C₂×C₄×C₅×C₁₇	C₂²×C₉	1	6	yes	—
39	C₂²×C₉×C₁₇	C₂×C₄×C₅	1	6	yes	—
40	C₂²×C₉×C₁₇	C₂³×C₅	1	1	yes	characteristic
41	C₄×C₉×C₁₇	C₂×C₄×C₅	1	12	yes	direct product factor
42	C₂×C₄²×C₁₇	C₅×C₉	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₃×C₅×C₁₇	C₃×C₄²	1	4	yes	—
44	C₂×C₃×C₅×C₁₇	C₂²×C₃×C₄	1	3	yes	—
45	C₂×C₃×C₄²×C₅	C₃×C₁₇	1	1	yes	characteristic
46	C₂³×C₃×C₁₇	C₂²×C₃×C₅	1	1	yes	characteristic
47	C₂×C₃×C₄×C₁₇	C₃×C₄×C₅	1	12	yes	—
48	C₂×C₃×C₄×C₁₇	C₂²×C₃×C₅	1	6	yes	—
49	C₂³×C₅×C₉	C₂²×C₁₇	1	1	yes	characteristic
50	C₂×C₄×C₅×C₉	C₄×C₁₇	1	12	yes	direct product factor
51	C₂×C₄×C₅×C₉	C₂²×C₁₇	1	6	yes	—
52	C₂²×C₅×C₁₇	C₂×C₄×C₉	1	6	yes	—
53	C₂²×C₅×C₁₇	C₂³×C₉	1	1	yes	characteristic
54	C₄×C₅×C₁₇	C₂×C₄×C₉	1	12	yes	direct product factor
55	C₂×C₉×C₁₇	C₄²×C₅	1	4	yes	direct product factor
56	C₂×C₉×C₁₇	C₂²×C₄×C₅	1	3	yes	—
57	C₂×C₄²×C₉	C₅×C₁₇	1	1	yes	characteristic, direct product factor, Hall
58	C₂²×C₄×C₁₇	C₂×C₅×C₉	1	3	yes	—
59	C₄²×C₁₇	C₂×C₅×C₉	1	4	yes	direct product factor
60	C₃×C₅×C₁₇	C₂×C₃×C₄²	1	1	yes	characteristic
61	C₂²×C₃×C₄×C₅	C₂×C₃×C₁₇	1	3	yes	—
62	C₃×C₄²×C₅	C₂×C₃×C₁₇	1	4	yes	—
63	C₂²×C₃×C₁₇	C₂×C₃×C₄×C₅	1	6	yes	—
64	C₂²×C₃×C₁₇	C₂³×C₃×C₅	1	1	yes	characteristic
65	C₃×C₄×C₁₇	C₂×C₃×C₄×C₅	1	12	yes	—
66	C₂²×C₅×C₉	C₂×C₄×C₁₇	1	6	yes	—
67	C₂²×C₅×C₉	C₂³×C₁₇	1	1	yes	characteristic
68	C₄×C₅×C₉	C₂×C₄×C₁₇	1	12	yes	direct product factor
69	C₂×C₅×C₁₇	C₄²×C₉	1	4	yes	direct product factor
70	C₂×C₅×C₁₇	C₂²×C₄×C₉	1	3	yes	—
71	C₂×C₄²×C₅	C₉×C₁₇	1	1	yes	characteristic, direct product factor, Hall
72	C₉×C₁₇	C₂×C₄²×C₅	1	1	yes	characteristic, direct product factor, Hall
73	C₂²×C₄×C₉	C₂×C₅×C₁₇	1	3	yes	—
74	C₄²×C₉	C₂×C₅×C₁₇	1	4	yes	direct product factor
75	C₂³×C₁₇	C₂²×C₅×C₉	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄×C₅×C₉	1	12	yes	direct product factor
77	C₂×C₄×C₁₇	C₂²×C₅×C₉	1	6	yes	—
78	C₂³×C₃×C₅	C₂²×C₃×C₁₇	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₃×C₄×C₁₇	1	12	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₃×C₁₇	1	6	yes	—
81	C₂×C₃×C₁₇	C₃×C₄²×C₅	1	4	yes	—
82	C₂×C₃×C₁₇	C₂²×C₃×C₄×C₅	1	3	yes	—
83	C₂×C₃×C₄²	C₃×C₅×C₁₇	1	1	yes	characteristic
84	C₂×C₅×C₉	C₄²×C₁₇	1	4	yes	direct product factor
85	C₂×C₅×C₉	C₂²×C₄×C₁₇	1	3	yes	—
86	C₅×C₁₇	C₂×C₄²×C₉	1	1	yes	characteristic, direct product factor, Hall
87	C₂²×C₄×C₅	C₂×C₉×C₁₇	1	3	yes	—
88	C₄²×C₅	C₂×C₉×C₁₇	1	4	yes	direct product factor
89	C₂³×C₉	C₂²×C₅×C₁₇	1	1	yes	characteristic
90	C₂×C₄×C₉	C₄×C₅×C₁₇	1	12	yes	direct product factor
91	C₂×C₄×C₉	C₂²×C₅×C₁₇	1	6	yes	—
92	C₂²×C₁₇	C₂×C₄×C₅×C₉	1	6	yes	—
93	C₂²×C₁₇	C₂³×C₅×C₉	1	1	yes	characteristic
94	C₄×C₁₇	C₂×C₄×C₅×C₉	1	12	yes	direct product factor
95	C₂²×C₃×C₅	C₂×C₃×C₄×C₁₇	1	6	yes	—
96	C₂²×C₃×C₅	C₂³×C₃×C₁₇	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂×C₃×C₄×C₁₇	1	12	yes	—
98	C₃×C₁₇	C₂×C₃×C₄²×C₅	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×C₃×C₅×C₁₇	1	3	yes	—
100	C₃×C₄²	C₂×C₃×C₅×C₁₇	1	4	yes	—
101	C₅×C₉	C₂×C₄²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
102	C₂³×C₅	C₂²×C₉×C₁₇	1	1	yes	characteristic
103	C₂×C₄×C₅	C₄×C₉×C₁₇	1	12	yes	direct product factor
104	C₂×C₄×C₅	C₂²×C₉×C₁₇	1	6	yes	—
105	C₂²×C₉	C₂×C₄×C₅×C₁₇	1	6	yes	—
106	C₂²×C₉	C₂³×C₅×C₁₇	1	1	yes	characteristic
107	C₄×C₉	C₂×C₄×C₅×C₁₇	1	12	yes	direct product factor
108	C₂×C₁₇	C₄²×C₅×C₉	1	4	yes	direct product factor
109	C₂×C₁₇	C₂²×C₄×C₅×C₉	1	3	yes	—
110	C₂×C₄²	C₅×C₉×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×C₃×C₅	C₃×C₄²×C₁₇	1	4	yes	—
112	C₂×C₃×C₅	C₂²×C₃×C₄×C₁₇	1	3	yes	—
113	C₂³×C₃	C₂²×C₃×C₅×C₁₇	1	1	yes	characteristic
114	C₂×C₃×C₄	C₃×C₄×C₅×C₁₇	1	12	yes	—
115	C₂×C₃×C₄	C₂²×C₃×C₅×C₁₇	1	6	yes	—
116	C₂²×C₅	C₂×C₄×C₉×C₁₇	1	6	yes	—
117	C₂²×C₅	C₂³×C₉×C₁₇	1	1	yes	characteristic
118	C₄×C₅	C₂×C₄×C₉×C₁₇	1	12	yes	direct product factor
119	C₂×C₉	C₄²×C₅×C₁₇	1	4	yes	direct product factor
120	C₂×C₉	C₂²×C₄×C₅×C₁₇	1	3	yes	—
121	C₁₇	C₂×C₄²×C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×C₄	C₂×C₅×C₉×C₁₇	1	3	yes	—
123	C₄²	C₂×C₅×C₉×C₁₇	1	4	yes	direct product factor
124	C₃×C₅	C₂×C₃×C₄²×C₁₇	1	1	yes	characteristic
125	C₂²×C₃	C₂×C₃×C₄×C₅×C₁₇	1	6	yes	—
126	C₂²×C₃	C₂³×C₃×C₅×C₁₇	1	1	yes	characteristic
127	C₃×C₄	C₂×C₃×C₄×C₅×C₁₇	1	12	yes	—
128	C₂×C₅	C₄²×C₉×C₁₇	1	4	yes	direct product factor
129	C₂×C₅	C₂²×C₄×C₉×C₁₇	1	3	yes	—
130	C₉	C₂×C₄²×C₅×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³	C₂²×C₅×C₉×C₁₇	1	1	yes	characteristic
132	C₂×C₄	C₄×C₅×C₉×C₁₇	1	12	yes	direct product factor
133	C₂×C₄	C₂²×C₅×C₉×C₁₇	1	6	yes	—
134	C₂×C₃	C₃×C₄²×C₅×C₁₇	1	4	yes	—
135	C₂×C₃	C₂²×C₃×C₄×C₅×C₁₇	1	3	yes	—
136	C₅	C₂×C₄²×C₉×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²	C₂×C₄×C₅×C₉×C₁₇	1	6	yes	—
138	C₂²	C₂³×C₅×C₉×C₁₇	1	1	yes	characteristic
139	C₄	C₂×C₄×C₅×C₉×C₁₇	1	12	yes	direct product factor
140	C₃	C₂×C₃×C₄²×C₅×C₁₇	1	1	yes	characteristic
141	C₂	C₄²×C₅×C₉×C₁₇	1	4	yes	direct product factor
142	C₂	C₂²×C₄×C₅×C₉×C₁₇	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₃	C₂³×C₃×C₅×C₁₇
socle, prime radical	C₂³×C₃×C₅×C₁₇	C₂²×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄²×C₅×C₉×C₁₇	C₂	direct (N × H)
5. C₂×C₄×C₅×C₉×C₁₇	C₄	direct (N × H)
7. C₂×C₄²×C₉×C₁₇	C₅	direct (N × H)
10. C₂×C₄²×C₅×C₁₇	C₉	direct (N × H)
12. C₄²×C₉×C₁₇	C₂×C₅	direct (N × H)
18. C₄×C₅×C₉×C₁₇	C₂×C₄	direct (N × H)
20. C₂×C₅×C₉×C₁₇	C₄²	direct (N × H)
22. C₂×C₄²×C₅×C₉	C₁₇	direct (N × H)
24. C₄²×C₅×C₁₇	C₂×C₉	direct (N × H)
26. C₂×C₄×C₉×C₁₇	C₄×C₅	direct (N × H)
33. C₅×C₉×C₁₇	C₂×C₄²	direct (N × H)
35. C₄²×C₅×C₉	C₂×C₁₇	direct (N × H)
37. C₂×C₄×C₅×C₁₇	C₄×C₉	direct (N × H)
41. C₄×C₉×C₁₇	C₂×C₄×C₅	direct (N × H)
42. C₂×C₄²×C₁₇	C₅×C₉	direct (N × H)
50. C₂×C₄×C₅×C₉	C₄×C₁₇	direct (N × H)
54. C₄×C₅×C₁₇	C₂×C₄×C₉	direct (N × H)
55. C₂×C₉×C₁₇	C₄²×C₅	direct (N × H)
57. C₂×C₄²×C₉	C₅×C₁₇	direct (N × H)
59. C₄²×C₁₇	C₂×C₅×C₉	direct (N × H)
68. C₄×C₅×C₉	C₂×C₄×C₁₇	direct (N × H)
69. C₂×C₅×C₁₇	C₄²×C₉	direct (N × H)
71. C₂×C₄²×C₅	C₉×C₁₇	direct (N × H)
72. C₉×C₁₇	C₂×C₄²×C₅	direct (N × H)
74. C₄²×C₉	C₂×C₅×C₁₇	direct (N × H)
76. C₂×C₄×C₁₇	C₄×C₅×C₉	direct (N × H)
84. C₂×C₅×C₉	C₄²×C₁₇	direct (N × H)
86. C₅×C₁₇	C₂×C₄²×C₉	direct (N × H)
88. C₄²×C₅	C₂×C₉×C₁₇	direct (N × H)
90. C₂×C₄×C₉	C₄×C₅×C₁₇	direct (N × H)
94. C₄×C₁₇	C₂×C₄×C₅×C₉	direct (N × H)
101. C₅×C₉	C₂×C₄²×C₁₇	direct (N × H)
103. C₂×C₄×C₅	C₄×C₉×C₁₇	direct (N × H)
107. C₄×C₉	C₂×C₄×C₅×C₁₇	direct (N × H)
108. C₂×C₁₇	C₄²×C₅×C₉	direct (N × H)
110. C₂×C₄²	C₅×C₉×C₁₇	direct (N × H)
118. C₄×C₅	C₂×C₄×C₉×C₁₇	direct (N × H)
119. C₂×C₉	C₄²×C₅×C₁₇	direct (N × H)
121. C₁₇	C₂×C₄²×C₅×C₉	direct (N × H)
123. C₄²	C₂×C₅×C₉×C₁₇	direct (N × H)
128. C₂×C₅	C₄²×C₉×C₁₇	direct (N × H)
130. C₉	C₂×C₄²×C₅×C₁₇	direct (N × H)
132. C₂×C₄	C₄×C₅×C₉×C₁₇	direct (N × H)
136. C₅	C₂×C₄²×C₉×C₁₇	direct (N × H)
139. C₄	C₂×C₄×C₅×C₉×C₁₇	direct (N × H)
141. C₂	C₄²×C₅×C₉×C₁₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₅×C₉×C₁₇	2
C₄²×C₅×C₉×C₁₇	2
C₂×C₃×C₄²×C₅×C₁₇	3
C₂×C₄²×C₉×C₁₇	5
C₂×C₄²×C₅×C₉	17

Subgroups — C2×C42×C5×C9×C17 #72201750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₅×C₉×C₁₇ #72201750